Positive Plücker tree certificates for non-realizability

Abstract

We introduce a new method for finding a non-realizability certificate of a simplicial sphere Σ: we exhibit a monomial combination of classical 3-term Plücker relations that yields a sum of products of determinants that are known to be positive in any realization of Σ; but their sum should vanish, contradiction. Using this technique, we prove for the first time the non-realizability of a balanced 2-neighborly 3-sphere constructed by Zheng, a family of highly neighborly centrally symmetric spheres constructed by Novik and Zheng, and several combinatorial prismatoids introduced by Criado and Santos. The method in fact works for orientable pseudo-manifolds, not just for spheres.

KEYWORDS:

• final polynomial
• realizability
• integer programming
• simplicial sphere
• Plücker relations

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Acknowledgments

It is a pleasure to thank Francisco Santos, Michael Joswig and Günter M. Ziegler for crucial discussions and their careful reading and pertinent suggestions. Moreover, I am very grateful to Amy Wiebe and Antonio Macchia for finding and pointing out an error in a previous version, to Isabella Novik and Hailun Zheng for stimulating discussions, and for pointing out various consequences of this work to their families of cs-neighborly spheres, and to the two anonymous referees for their pertinent input.

Additional information

Funding

The author was supported by the grant PID2019-106188GB-I00 from the Spanish Ministry of Education (MEC)